Abstract: In the paper Quantum flag varieties, equivariant quantum -modules, and localization of Quantum groups, Backelin and Kremnizer defined categories of equivariant quantum -modules and -modules on the quantum flag variety of . We proved that the Beilinson-Bernstein localization theorem holds at a generic . Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between categories of -modules and -modules on the quantum flag variety.

For this we first prove that is an Azumaya algebra over a dense subset of the cotangent bundle of the classical (char 0) flag variety . This way we get a derived equivalence between representations of and certain -modules.

In the paper Localization for a semi-simple Lie algebra in prime characteristic, by Bezrukavnikov, Mirkovic, and Rumynin, similar results were obtained for a Lie algebra in char . Hence, representations of and of (when is a 'th root of unity) are related via the cotangent bundles in char 0 and in char , respectively.